Q. If $Δ = ∣ ∣ 111 x y z x^{2} y^{2} z^{2} ∣ ∣$ and $Δ_{1} = ∣ ∣ 1 yz x 1 z x y 1 x y z ∣ ∣$ , then

165 150 Determinants Report Error

A

$Δ + Δ_{1} = 1$

B

$Δ - Δ_{1} = 0$

C

$Δ + Δ + 0$

D

$Δ - Δ_{1} = 1$

Solution:

We have, $Δ_{1} = ∣ ∣ 1 yz x 1 z x y 1 x y z ∣ ∣$
By interchanging rows and columns, we get
$Δ_{1} = ∣ ∣ 111 yz z x x y x y z ∣ ∣ = \frac{1}{x yz} ∣ ∣ x y z x yz x yz x yz x^{2} y^{2} z^{2} ∣ ∣ = \frac{x yz}{x yz} \cdot ∣ ∣ x y z 111 x^{2} y^{2} z^{2} ∣ ∣$
$= (- 1) ∣ ∣ 111 x y z x^{2} y^{2} z^{2} ∣ ∣ = - Δ$ (interchanging $C_{1}$ and $C_{2})$
$\Rightarrow Δ_{1} + Δ = 0$